Application of Rheological Models in Prediction of Turbulent Slurry ...

Flow Turbulence Combust (2010) 84:277–293 DOI 10.1007/s10494-009-9234-y

Application of Rheological Models in Prediction of Turbulent Slurry Flow Artur Bartosik

Received: 11 August 2008 / Accepted: 8 September 2009 / Published online: 20 September 2009 © Springer Science + Business Media B.V. 2009

Abstract The paper deals with fully developed steady turbulent flow of slurry in a circular straight and smooth pipe. The Kaolin slurry consists of very fine solid particles, so the solid particles concentration, and density, and viscosity are assumed to be constant across the pipe. The mathematical model is based on the time averaged momentum equation. The problem of closure was solved by the Launder and Sharma k-ε turbulence model (Launder and Sharma, Lett Heat Mass Transf 1:131–138, 1974) but with a different turbulence damping function. The turbulence damping function, used in the mathematical model in the present paper, is that proposed by Bartosik (1997). The mathematical model uses the apparent viscosity concept and the apparent viscosity was calculated using two- and three-parameter rheological models, namely Bingham and Herschel–Bulkley. The main aim of the paper is to compare measurements and predictions of the frictional head loss and velocity distribution, taking into account two- and three-parameter rheological models, namely Bingham and Herschel–Bulkley, if the Kaolin slurry possesses low, moderate, and high yield stress. Predictions compared with measurements show an observable advantage of the Herschel–Bulkley rheological model over the Bingham model particularly if the bulk velocity decreases. Keywords Non-Newtonian flow · Prediction of turbulent slurry flow · Turbulence damping function · Rheological models

A. Bartosik (B) Kielce University of Technology, Al. Tysiaclecia ˛ P.P. 7, 25-314 Kielce, Poland e-mail: [email protected]

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Nomenclature Cv Ci D d d50 fμ k K le n p r, x R Re S St t∗ te u , v U Ub Ue y

volumetric concentration of solid particles in carrier liquid, % constants in Launder and Sharma turbulence model, i = 1, 2 inner pipe diameter, mm particle diameter, median particle diameter, μm turbulence damping function, kinetic energy of turbulence, m2 /s2 coefficient in Herschel–Bulkley rheological model, Pa sn characteristic length of swirl, m power exponent in Herschel–Bulkley rheological model, static pressure, Pa distance from symmetry axis/axial coordinate, m inner pipe radius, m Reynolds number, ratio of solid/liquid density, Stokes number, relaxation/response time, s time of appearance of micro-scale swirls, s fluctuating components of velocity, m/s slurry velocity, m/s bulk velocity, m/s characteristic velocity of swirl, m/s distance from the pipe wall, m

Greek Symbols γ˙ ε μ ρ σk σε τ, τ0

strain rate (shear deformation rate), 1/s rate of dissipation of kinetic energy of turbulence, m2 /s3 viscosity, Pa s density, kg/m3 effective Prandtl-Schmidt number for k, effective Prandtl-Schmidt number for ε, shear stress/yield stress, Pa

Subscripts ap b ef L m PL S t w

apparent viscosity bulk effective liquid Kaolin slurry (solid–liquid mixture) plastic solid turbulent/eddy wall


279

1 Introduction Turbulent flow of slurries with a fine dispersive solid phase at high volumetric concentration and with a yield stress appears in a variety of industrial processes such as transportation in power engineering, chemical engineering, waste-disposal industry, mining, etc. While a pipeline is undoubtedly the most efficient method for transporting slurries, its efficiency depends upon two fundamental requirements: it must operate continuously, and the required throughput must be obtained with the least capital investment and the lowest operating costs. The effective design, control, and operation of a slurry transport system, composed of a pipeline and pumps, requires the successful prediction of frictional head loss. Developing a mathematical model to predict slurry frictional head loss always requires an answer to the fundamental question as to whether the flow is homogeneous, pseudo-homogeneous, or heterogeneous. The flow patterns depend on several parameters, for example: particle sizes, particle density, slurry density, slurry viscosity, slurry velocity, particle shape, slip velocity, etc. Taking into account the slip velocity, laser measurements of Nouri et al. [1] conducted for fully developed turbulent pipe flow of solid–liquid mixture with particles of mean diameter ranging between 270 μm and a few microns, indicated no significant differences in the liquid and solid phase velocities. They found that the slip velocity decrease with the solid particles concentration increase. However, as a result of experimental difficulties, the maximum solid particles concentration they used their experiments was 14% by volume. The authors cited earlier observations of other researchers who also found that for similar particle diameters and for fully developed flow there was no substantial slip between the particles and the liquid. Concluding one can say that the importance of the slip velocity exists in dilute solutions and in case of moderate and high solid particles concentration can be neglected. Taking into account the particle sizes the measurements of Nasr-El-Din et al. [2] and Sumner et al. [3] indicated a strong influence of the particle diameter on solid-phase concentration distribution across the pipe, particularly near the pipe wall. They reported that for fine, medium, and coarse sand particles the concentration is not uniform, so the flow is not homogeneous. They observed that with the particle diameter increase the solid-phase concentration decreases at the pipe wall. In aim to predict the frictional head loss in slurry flow with medium, and coarse particles, Shook and Roco [4] suggested, that it is reasonable to assume the Newtonian model for the shear stress. On this basis the frictional head loss can be predicted using density of the slurry and the viscosity equal to the carrier liquid phase. On the basis of the solid particles concentration measurements, performed by Nasr-El-Din et al. [2] and Sumner et al. [3], and taking into account knowledge gained from Nouri et al. [1], Bartosik and Shook [5, 6] used measured solid-phase concentration distribution data and one- and two-equation turbulence models to predict frictional head loss for fully developed turbulent vertical pipe flow in the range of solid particles concentration CV = (10–40)%. The numerical predictions were carried out for the mixture of water, and fine, and medium, and coarse sand with mean solid particle diameters d = 175 μm, d = 470 μm, and d = 780 μm, respectively. The authors made the simplification in the mathematical model that the presence of the particles is almost neutral to turbulence compared to a

280


single-phase flow with the same flow conditions, so there was no substantial increase or decrease of the turbulence level compared to a single-phase flow with the same flow conditions. They calculated the local slurry density and the local slurry viscosity from the measured solid particles concentration distribution taking into account Thomas [7] equation for viscosity. Their results of predicted frictional head loss and local velocity distribution for slurry with fine, and median particle diameter confirmed that it is possible to successfully extend the application of the standard turbulence model. However, authors noted that the k-ε model, proposed by Launder and Sharma [8], gave the best results compared to Hassid-Poreh [9], Jones-Launder [10], and Chien [11] turbulence models. For coarse particles, however, particlesfluid and particle-particle interactions play important roles, so the results of the predictions were not so satisfactory. For the slurry flow with coarse particles, a model which includes particle–wall and particle–particle interactions was proposed by Bagnold [12] and developed by Shook and Bartosik [13], and Bartosik [14]. In the papers of Hetsroni et al. [15, 16], and Kaftori et al. [17], we can find a wide review of literature on the interactions between solid particles and the carrier fluid including a review of turbulence models and matching of experimental data. However, as a result of experimental difficulties in solid–liquid flows, especially at the pipe wall, the majority of the mathematical models are dedicated to liquid–gas or solid–gas flows. The effect of particle size and particle shape on pipeline frictional head loss can be found in the papers of Shaan et al. [18], Kitanovski and Poredos [19], and Kaushal et al. [20]. Concluding, one can say that the presence of the solid particles may substantially affect turbulence parameters and transport properties. Depending on the particle size, particles density and the solid particles concentration the particles can suppress or enhance the level of turbulence or in some cases can be almost neutral for the turbulence, compared to a single-phase flow with the same flow conditions. However, no simple parameter has been defined so far, which could indicate if there is or is not an increase or suppression of the turbulence in the solid– liquid flow. d2S ρS 18 μL

(1.1)

te =

le Ue

(1.2)

St =

t∗ te

(1.3)

t∗ =

If slurry with median particle diameter of a few microns appears, it is common that the slurry exhibits non-Newtonian behaviour with the yield stress. In turbulent slurry flow with such fine particles the solid particles concentration is constant across the pipe as a result that particles are small enough, so they are uniformly distributed by turbulent diffusion. Taking into account relaxation time of solid particles (t∗ ), defined by (1.1), and Kolmogorov time scale (te ), defined by (1.2), one can say that if the Stokes number (1.3) is below unity (St < 1) the particles can follow a carrier


281

liquid motion. The relaxation time of the Kaolin particles used in the present work is very low, and for d50 = 5 μm, and solid particles density ρS = 2,650 kg/m3 , submerged in a carrier liquid with viscosity μL = 10.046 10−4 Pa s, is equal t∗ = 3.7 μs. If the relaxation time is higher than the time of small scale swirls (te ), so the St > 1, the solids will not follow a carrier liquid motion causing that slip velocity and/or turbulence generation appears. The numerical predictions of the slurry flow with very fine particles were proposed by Stainsby and Chilton [21]. They used a hybrid model assuming that the rheology of the fluid with the yield stress can be represented by the Herschel–Bulkley model at the low strain rate and the Bingham plastic model at the high strain rate. Instead of molecular viscosity they used apparent viscosity. The hybrid model required five parameters which are derived from the viscometer data. They used a one dimensional finite difference model [22], which solves the time-averaged momentum equation for fully developed flow in a circular pipe. The closure was performed be means of the Launder and Sharma [8] turbulence model. They did not introduce any changes into the turbulence model. Predictions have been in satisfactory agreement with experimental data. However, it must be noted that prediction of frictional head loss has been verified only for very low yield stress and low plastic viscosity, which is always easier for successful prediction. Prediction of frictional head loss of Kaolin slurry, for moderate and high yield stress, using the standard turbulence model is unsatisfactory as a result of high discrepancy with measurements. Predicted frictional head loss is much higher, compared to experimental data. This is due to the fact that the fine dispersive phase modifies the viscous sub-layer and buffer layer causing appearance of damping of the turbulence nears the wall. For such slurries the hypothesis proposed by Wilson and Thomas is most useful [23]. Wilson and Thomas reasoned that in the slurry flow with very fine solid particles the viscous sub-layer becomes thicker. It means that turbulence in the near-wall region is suppressed compared to a single-phase flow with the same flow conditions. Using their hypothesis these authors developed an algebraic model for prediction of frictional head loss. The authors have extended their concept recently to prediction of transition velocity [24], and from smooth to rough walled pipes [25]. As a result of difficulties in obtaining experimental data of turbulence near the pipe wall for the Kaolin slurry with moderate and high solid particles concentration it is impossible to propose a new turbulence model. So, taking into account the Wilson and Thomas hypothesis, Bartosik [26] modified Launder and Sharma turbulence damping function by taking into account a dimensionless yield stress (τo /τw ). The modified turbulence damping function approaches the standard Launder and Sharma one as the yield stress approaches zero. As a consequence of introducing the modified turbulence damping function a reduction of turbulence stress component (u v ) at the wall takes place. Finally, the time-averaged momentum equation, and ‘k-ε’ turbulence model with the modified turbulence damping function compose the mathematical model. Such a mathematical model together with the Bingham rheological model, determining the apparent viscosity, was examined for a comprehensive range of rheological parameters, and for a few pipe diameters, confirming good agreement with measurements [27, 28]. Recently, the mathematical model was extended by using the three-parameter Herschel–Bulkley rheological model giving good results again [29].

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Taking into account a slurry flow with very fine particles it is always questionable which rheological model is most suitable, such as two-parameter, like Bingham, which is commonly used in engineering application, or three-parameter, like Herschel–Bulkley, which better describes the shear stress for the low strain rate. So, the main aim of the paper is to examine application of two- and threeparameter rheological model, namely Bingham and Herschel–Bulkley, together with ‘k-ε’ turbulence model, which includes modified turbulence damping function, for predictions of frictional head loss and velocity distribution if the Kaolin slurry possesses low, moderate, and high yield stress. The availability of the mathematical model and appropriate rheological model has made it possible to conduct design of a pipelines characteristics and optimization studies through numerical simulation.

2 Mathematical Model This paper deals with fully developed steady turbulent flow in a circular horizontal and smooth pipe. The flow is axially symmetrical without a circumferential velocity component. The Kaolin slurry consists of solid particles, with mean particle diameter of a few microns, so the solid particles concentration is constant across the pipe and the flow is assumed to be homogenous with constant density and viscosity across the pipe. The mathematical model, for the aforementioned assumption, is composed in cylindrical co-ordinates using time-averaged momentum equation, described as: 1 ∂ ∂U ∂p r μ − ρ u v (2.1) = r ∂r ∂r ∂x in which component of the turbulent stress tensor, designated using Boussinesq hypothesis, can be written as follows: -ρu v = μt

∂U ∂r

(2.2)

The closure of the momentum equation (2.1) is solved by using Launder and Sharma turbulence model [8], which is described as follows: Kinetic energy of turbulence: 2 2 μt ∂k 1 ∂U ∂k1/2 r μ+ + μt = ρε + 2μ r σk ∂r ∂r ∂r

(2.3)

Dissipation rate of kinetic energy of turbulence: 2 μt ∂ε ε 1 ∂U r μ+ + C1 μt r σε ∂r k ∂r

ρε2 − 2νμt = C2 1 − 0.3 exp −Re2t k

∂ 2U ∂r2

2 (2.4)


283

Turbulent viscosity: μt = fμ Turbulence damping function:

ρ k2 ε

fμ = 0.09 EXP

−3.4

(2.5)

(1 + Ret /50)2

(2.6)

Turbulent Reynolds number: Ret =

ρ k2 εμ

(2.7)

Bartosik [26] proposed a modification of the turbulence damping function (2.6), by taking into account a dimensionless yield stress (τ0 /τw ). On the bases of comparison between predictions and measurements he proved that for the Kaolin slurry the turbulence damping function should be expressed as follows: ⎤ ⎡ τo −3.4 1 + ⎢ τw ⎥ (2.8) fμ = 0.09 EXP ⎢ 2 ⎥ ⎦ ⎣ 1 + Ret 50 The modified turbulence damping function, described by (2.8), approaches the standard damping function (2.6), as the yield stress approaches zero. However, with the yield stress increase, at the same flow conditions, the turbulence damping function decreases the turbulent viscosity (2.5) and in consequence decreases the turbulent shear stresses (2.2) at the pipe wall. This is coherent with the Wilson and Thomas hypothesis [23] who reasoned that the viscous sub-layer becomes thicker, compared to a single-phase flow at the same flow conditions, if slurry with very fine solid particles is considered. Taking into account Bingham and Herschel–Bulkley rheological models one can calculate the apparent viscosity, defined as: τ = μap γ

(2.9)

as follows: For Bingham rheological model: γ = 0 for τ ≤ τo

(2.10)

and τ = τo + μPL γ for τ > τo So, taking into account (2.9) and (2.11) the apparent viscosity for γ > 0 is: μ μap = PLτ 0 1− τ

(2.11)

(2.12)

For Herschel–Bulkley rheological model: γ = 0 for τ ≤ τo

(2.13)

284


and τ = τo + Kγn

for τ > τo

Substituting the strain rate from (2.9) to (2.14) we can get: n τ τ = τo + K μap So, τ = μap

τ − τo K

(2.14)

(2.15)

1/n

So, for γ > 0 the apparent viscosity was set up as follows: τ K1/n τ μap = 1/n = (τ − τo )1/n τ − τo

(2.16)

(2.17)

K Instead of the viscosity (μ) that appears in (2.1) and (2.3) and (2.4) the apparent viscosity (μap ) is used in the mathematical model. It must be noted however, that (2.12) and (2.17) have limitations. For both rheological models the yield stress cannot be equal to or exceed the shear stress. However, such situations can take place in the close vicinity of the symmetry axis. This is due to the fact that the shear stress varies linearly from its maximum value at the pipe wall to zero in the symmetry axis. It means that at some distance from the symmetry axis the shear stress can be lower than the yield stress. To avoid the situation where the yield stress can be higher or equal to the shear stress, it was assumed that the apparent viscosity is constant across the pipe and is the same as at the pipe wall. So, the apparent viscosity in the mathematical model was calculated for Bingham and Herschel–Bulkley rheological models as following: For Bingham rheological model: μap =

μPL τ0 1− τw

(2.18)

For Herschel–Bulkley rheological model: μap =

K1/n τw (τw − τo )1/n

(2.19)

The wall shear stress appeared in (2.18) and (2.19) was designated from balance of forces acting on unit pipe length, so the wall shear stress was calculated as: τw =

p D , x 4

(2.20)

The bulk Reynolds number was calculated using apparent viscosity, as following: Re =

ρm Ub D μap

(2.21)

Finally, the mathematical model was set up by three partial differential equations, namely (2.1), (2.3)–(2.4), together with the complimentary (2.2), (2.5), (2.7), (2.8) and


285

the apparent viscosity (2.18) and (2.19), which corresponds respectively to Bingham and Herschel–Bulkley rheological models. All numerical calculations were made for known p/x. The mathematical model assumes non slip velocity at the pipe wall, i.e. U = 0, and k = 0, ε = 0. Axially symmetrical conditions were applied at the pipe centre to all variables with dU/dr = 0, dk/dr = 0 and dε/dr = 0. The turbulence constants in the model are the same as those in the Launder and Sharma model: C1 = 1.44; C2 = 1.92; σk = 1.0; σε =1.3. The mathematical model was solved by finite difference scheme using own computer code. Differential grid of 80 nodal points distributed on the radius of the pipe has been used. Majority of the nodal points were localised near the pipe wall. The number of nodal points was set experimentally to provide nodally independent predictions. The mathematical model was examined for a comprehensive range of rheological parameters defined by Bingham, Casson, and Herschel–Bulkley models, and for a few pipe diameters confirming satisfying agreement of predictions with measurements, Bartosik [26–29].

3 Results of Prediction Numerical predictions have been performed for fully developed steady turbulent flow in a circular horizontal and smooth pipe. The slurry flow consists of very fine solid particles, in which median particle diameter was a few microns, and the slurry density and the apparent viscosity were assumed to be a constant across the pipe. Predictions, using two- and three-parameter rheological models like Bingham and Herschel–Bulkley, were performed for the frictional head loss and the local velocity distribution. The predictions were compared with the literature, and the author’s own experimental data for the range of pipes diameter and rheological parameters shown in Table 1. For all sets of experiments a slurry temperature was about 25o C. It is worth to emphasis that the apparent viscosity, described by (2.18) and (2.19), which corresponds to the Bingham and the Herschel–Bulkley rheological model, depends on the yield stress, while the yield stress depends on the solid particles concentration. Dependence of the yield stress on the solid particles concentration is a physical property, which can differs for different slurries. As an example Fig. 1 illustrate relation τo = f(CV ) measured for Couette flow of Kaolin slurry with median particle diameter d50 = 10 μm [30]. As it is seen the relation is exponential. As an example Fig. 2 illustrate calculated relation of μap = f(τo ), using Bingham model, described by (2.18). The relation is also exponential. Predictions of the frictional head loss vs. bulk velocity for Kaolin slurry with low yield stress, based on the Bingham and the Herschel–Bulkley (H-B Model) rheological models, were compared with Slatter [31] experimental data—Fig. 3. Both

Table 1 Range of pipe diameters and rheological parameters of the Bingham and the Herschel– Bulkley rheological models used for comparison of predictions with measurements D (mm)

Cv %

ρm (kg/m3 )

τo (Pa)

μPL (Pa s)

K (Pa sn )

Re

159; 207

7.3–47.7

1,105–1,667

4.9–43.0

0.0043–0.05

0.035–0.113

40,800–147,000

286


Fig. 1 Measured yield stress for Kaolin slurry; d50 = 10 μm; S = 2.55

20

Yield Stress, Pa

18 16 14 12 10 8 6 4 2 0 15

17

19

21

23

CV, %

models gave similar agreements with measurements except for the low flow rate where the Bingham model under predicts the frictional head loss by almost 12%. Predictions of the frictional head loss based on the Herschel–Bulkley model are in very good agreement with the experimental data of Slatter [31]. The range of Reynolds number, defined by (2.21), which is Re = (40,830–110,740), corresponds to the lowest and the highest bulk velocity predicted using Bingham rheological model (Fig. 3). Additionally, Fig. 3 includes data for the water flow in a smooth pipe (dashed line with points), and results of numerical predictions using standard turbulence damping function (SDF), described by (2.6), together with the apparent viscosity calculated using Bingham model (2.18). As it is seen predictions performed using standard turbulence damping function (SDF), shown as dashed line, gave much higher dp/dx values compared to experimental data, and to predictions performed using modified turbulence damping function (points), described by (2.8). For this case the advantage of the mathematical model, which includes modified damping function (2.8), is substantial. For moderate value of the yield stress, comparison of predictions with measurements of Xu et al. [32] gave similar results as before, and again small discrepancies between the Bingham and the Herschel–Bulkley models occurred for low bulk velocities, as shown in Fig. 4. The relative error for the Bingham model does not

0.35

Apparent Viscosity, Pa s

Fig. 2 Calculated apparent viscosity for Kaolin slurry using Bingham Model; D = 159 mm; S = 2.4; ρm = 1,667 kg/m3 ; Cv = 46.4%; μPL = 0.05 Pa s; dp/dx = 1,500 Pa/m

0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

10

20

30

Yield Stress, Pa

40

50

Flow Turbulence Combust (2010) 84:277–293 350 300

dp/dx, Pa/m

Fig. 3 Predicted and measured frictional head loss for Kaolin slurry; D = 207 mm; ρm = 1,105.3 kg/m3 ; Cv = 7.4%; S = 2.44; Bingham Model: τo = 4.924 Pa; μPL = 0.00433 Pa s; Re = (40,830–110,740); H-B Model: τo = 4.18 Pa; K = 0.03509 Pa sn ; n = 0.719

287

250 200 Exp. Slatter H- B Model Bingham Model SDF Water

150 100 50 0 1.5

2.0

2.5

3.0

3.5

Bulk Velocity, m/s

exceed 10% and again the Herschel–Bulkley model gave very good agreement with measurements. For the set of parameters shown in Fig. 4 there is no significant difference between numerical predictions using standard (dashed line) and modified turbulence damping function (points). Figure 4 illustrate that the laminar-turbulent transition appears aproximately for Ub ≈ 1.6 m/s. For high yield stress and low bulk velocities, both rheological models under predict frictional head loss and the relative error for both models is about 12%, as shown in Fig. 5. The accuracy of the Herschel–Bulkley model increases with the bulk velocity increase, while the Bingham model under predicts frictional head loss over the whole range of measured bulk velocities. The relative error for the Bingham model varies from 12% for the low bulk velocity up to 5% for the high bulk velocity, while the Herschel–Bulkley model respectively from 12% to almost 0%. However, in spite of the difficulties in predictions of Kaolin slurry flow for the high yield stress the results of prediction can be treated as fully acceptable. Figure 5 confirme that results of numerical predictions performed using modified damping function (2.8) compared to standard damping function (SDF) are substanially different and SDF is not appropriate to poredict Kaolin slurry flow. It is seen as well that the laminar flow appears aproximately for Ub < 3.8 m/s. In case of Kaolin slurry flow the laminar–turbulent transition take place at higher critical Reynolds number than we expect [30].

2500 2000

dp/dx, Pa/m

Fig. 4 Predicted and measured frictional head loss for Kaolin slurry; D = 159 mm; ρm = 1,535 kg/m3 ; S = 2.4; Cv = 38.3%; Bingham Model: τo = 9 Pa; μPL = 0.013 Pa s; Re = (23,210–84,860); H-B Model: τo = 4.5 Pa; K = 0.054 Pa sn ; n = 0.84

1500

Exp. Xu et al. Bingham Model H-B Model SDF Water

1000 500 0 0.0

1.0

2.0

3.0

Bulk Velocity, m/s

4.0

5.0

288 3000 2500

dp/dx, Pa/m

Fig. 5 Predicted and measured frictional head loss for Kaolin slurry; D = 159 mm; ρm = 1,667 kg/m3 ; S = 2.4; Cv = 47.7%; Bingham Model: τo = 43 Pa; μPL = 0.05 Pa s; Re = (8,210–14,890); H-B Model: τo = 39 Pa; K = 0.113 Pa sn ; n = 0.89


2000

Exp. Xu et al. Bingham Model H-B Model SDF Water

1500 1000 500 0 0.0

1.0

2.0

3.0

4.0

5.0

Bulk velocity, m/s

Considering kinematics of the Kaolin slurry flow it is interesting to compare measured and predicted velocity profiles using the Bingham and the Herschel–Bulkey rheological models. Figure 6 shows predicted and measured velocity profile for the moderate yield stress of Kaolin slurry. Predictions were compared with experimental data of Xu et al. [32]. As it is seen both models gave almost the same quantitative and qualitative velocity shape and fit experimental data well. Additionally, Fig. 6 presents velocity profile of water. Comparison of velocity profiles of water and Kaolin slurry indicates some differences at the pipe wall. The velocity profile of single-phase Newtonian liquid is significantly steeper at the pipe wall compared to two-phase nonNewtonian slurry. It must be noted however, that to see eventual differences in the velocity shape the predictions using both rheological models were made for the same velocity in symmetry axis. The same quantitative and qualitative agreement between predicted and measured velocity profiles, as for moderate yield stress, was noted for high yield stress. For high yield stress, predicted velocity profiles using the Bingham and the Herschel– Bulkley rheological models were compared with measurements of Xu et al. [32] and are presented in Fig. 7. Both rheological models showed again good agreement with measurements, except for the measured point closest at the pipe wall. It is difficult to undoubtly state however, that the predictions of the velocity distribution fail at the

5.0 4.5

Local Velocity, m/s

Fig. 6 Predicted and measured velocity distribution for Kaolin slurry; D = 159 mm; ρm = 1,535 kg/m3 ; S = 2.4; Cv = 38.3%; Ub = 3.5 m/s; Bingham Model: τo = 9 Pa; μPL = 0.013 Pa s; Re = 52,970; H-B Model: τo = 4.5 Pa; K = 0.054 Pa sn ; n = 0.84

4.0 3.5 3.0 2.5 2.0

Exp. Xu et al. Bingham Model

1.5

H-B Model Water

1.0 0.5 0.0 0.0

0.2

0.4

0.6

y/R

0.8

1.0

Flow Turbulence Combust (2010) 84:277–293 6.0 5.0

Local Velocity, m/s

Fig. 7 Predicted and measured velocity distribution for Kaolin slurry and water; D = 159 mm; S = 2.4; ρm = 1,667 kg/m3 ; Cv = 47.7%; Ub = 4.5 m/s; Bingham Model: τo = 43 Pa; μPL = 0.05 Pa s; Re = 11,770; H-B Model: τo = 39 Pa; K = 0.113 Pa sn ; n = 0.89

289

4.0 3.0 Eksp. Xu et al. Bingham Model H-B Model Water

2.0 1.0 0.0 0.0

0.1

0.2

0.3

0.4

0.5 y/R

0.6

0.7

0.8

0.9

1.0

pipe wall, because the measurements performed by Xu et al. [32] were conducted using a Pitot tube which means that measurements are always uncertain in the vicinity of the solid wall, especially when the solid particles concentration is high. Comparing velocity profiles of Kaolin slurry and water it is evident that the profiles are substantially different. The water velocity profile is significantly steeper at the pipe wall. τo τw ρm 1 − (R − r) ρm τw R+ = (3.1) μPL U U+ = τw ρm

(3.2)

Precicted and measured velocity profile made in logarithmic scale, where U+ and R+ are defined by (3.1) and (3.2), confirms that velocity distribution of Kaolin slurry and water are different, what is shown in Fig. 8. Again the Bingham and the Herschell-Bulkley model gave similar results and both demonstrateted good agreement with measurements exept point closest at the pipe wall as was discussed earlier. Analyzes of numerical simulation and experimental data of velocity profiles

35 30 25

U+

Fig. 8 Logaritmic velocity distribution for Kaolin slurry; D = 159 mm; ρm = 1,667 kg/m3 ; S = 2.4; Cv = 47.7%; Ub = 4.5 m/s; Re = 11,775; Bingham Model: τo = 43 Pa; μPL = 0.05 Pa s; H-B Model: τo = 39 Pa; K = 0.113 Pa sn ; n = 0.89

20 15 Eksp. Xu et al.

10

Bingham Model HBM Model

5 0 1E+00

Water

1E+01

1E+02

1E+03

R+

1E+04

1E+05

290


for Kaolin slurry indicates that with the yeild stress increase (increase of CV ) the velocity profile is changing qualitatively and quantitavely, compared to a single-phase of Newtonian liquid at the same flow conditions. As was emphasised earlier the velocity profile of a single-phase of Newtonian liquid is significantly steeper at the pipe wall. The velocity profile of Kaolin slurry demonstrate higher voscous forces at the pipe wall. Comparison of predictions of frictional headloss and velocity profiles for turbulent flow of Kaolin slurry shows that the Bingham and the Herschel–Bulkley models are useful for engineering application if ‘k-ε’ model with modified turbulence damping function, which includes the yield stress, is used. Numerical computations confirmed that both rheological model give satisfactory predictions for low, moderate, and high yield stresses. However, the Herschel–Bulkley model shows an advantage, compared to the Bingham model, if the bulk velocity is low due to the fact that the shear stress for the low strain rate is better described by the Herschel–Bulkley model. To demonstrate importance of the modification made in the damping function, described by (2.8), which enhance damping of turbulence in Kaolin slurry flow in close vicinity of the pipe wall, the Fig. 9 displays dimensionless shear stress across the pipe. The dimensionless shear stress is defined as ratio of turbulent shear stress to effective shear stress. The effective shear stress is treated as a sum of viscous and turbulent shear stresses and is described, as follows: τef = μap + μt γ˙ (3.3) So, the dimensionless shear stress is following: μt γ˙ 1 − ρ u v = = μ τef μap + μt γ˙ ap +1 μt

(3.4)

The dimensionless shear stress, displayed in Fig. 9, demonstrate influence of the yield stress on reduction of turbulent stress in the near wall region. When the yield stress increases, from zero up to τo = 30 Pa, the turbulence near the wall is suppressed, so the viscous sublayer becomes thicker and dominating forces are the viscous forces. The numerical prediction are coherent with the Wilson and Thomas

1.00 0.90 0.80 0.70

-ρu'v'/τef

Fig. 9 Dependence of dimensionless turbulent shear stress on R+ for Kaolin slurry; D = 159 mm; ρm = 1,535 kg/m3 ; S = 2.4; Cv = 38.3%; H-B Model: K = 0,075 Pa sn ; dp/dx = 1,500 Pa/m; n = 0,84

0.60 το=0 το=5 το=15 το=30

0.50 0.40 0.30 0.20 0.10 0.00 1

10

100

R+

1000


291

hypothesis who reasoned that for Kaolin slurry with the yield stress the viscous sublayer becomes thicker and the velocity profile would become less steep at the pipe wall, Wilson and Thomas [23].

4 Conclusions It is worth emphasizing that single-phase flow of liquid or gas has been investigated extensively for several years and information gained from experiments has contributed to the development of reliable turbulence models. The difficulties increase when two-phase flow is considered because of lack of reliable experimental data of turbulence in close vicinity of the solid wall, especially for moderate and high solids concentration. However, it is possible to suggest modification of turbulence models based on comparison between computations and measurements of global parameters, which are particularly relevant in engineering practice. It must be noted, however, that the damping function is simply a correction function and is quite arbitrary. Some of proposed turbulence models do not need turbulence damping function [33, 34]. Nevertheless, turbulence models with the damping function can be effective for engineering calculations. In the situation described in the present work the mathematical model was successfully examined for Kaolin slurry flow in comprehensive range of parameters giving satisfying predictions of frictional head loss and velocity profiles [26–29]. Recent comparison of experimental data of Kaolin slurry with prediction using the aforementioned mathematical model and with support of two-parameter rheological models, like Bingham and Casson, show that the rheological model is not a crucial factor [35]. The crucial factor in this case is Wilson and Thomas hypotheses [23] together with modification made in the turbulence damping function [26]. However, when two- or three-parameter rheological models are considered the importance of the model can play a role as a consequence that the three-parameter model better describes the shear stress at the low shear deformation rate. On the basis of comparison between measurements and predictions of Kaolin turbulent slurry flow, using ‘k-ε’ turbulence model with modified turbulence damping function, and taking into account the Bingham and the Herschel–Bulkley rheological models the following conclusions have been drawn: 1. For the low and moderate yield stress the Herschel–Bulkley rheological model gives very good agreement of predictions with measurements over a comprehensive range of bulk velocities, while the Bingham rheological model under predicts frictional head loss for low bulk velocities. 2. For the high yield stress both rheological models under predict frictional head loss for low bulk velocities. The discrepancy between predictions and measurements decreases with the bulk velocity increase and the Herschel–Bulkley model posses’ quantitative advantage compared to the Bingham model. 3. The velocity profile of Kaolin slurry flow is substantially different compared to a water flow and is significantly less steep at the pipe wall and can not be properly predicted if standard turbulence model is used. 4. Predictions of the velocity distribution by both rheological models show good agreement with measurements for moderate and high yield stress.

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5. For the slurry parameters used in the present work, there is an observable advantage of the Herschel–Bulkley rheological model over the Bingham model. The advantage is pronounced for the low bulk velocity which is due to the fact that the Herschel–Bulkley model better predicts the shear stress for the lower shear deformation rate.

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